Hopf bifurcations in dynamical systems

Abstract

The onset of instability in autonomous dynamical systems (ADS) of ordinary differential equations is investigated. Binary, ternary and quaternary ADS are taken into account. The stability frontier of the spectrum is analyzed. Conditions necessary and sufficient for the occurring of Hopf, Hopf–Steady, Double-Hopf and unsteady aperiodic bifurcations—in closed form—and conditions guaranteeing the absence of unsteady bifurcations via symmetrizability, are obtained. The continuous triopoly Cournot game of mathematical economy is taken into account and it is shown that the ternary ADS governing the Nash equilibrium stability, is symmetrizable. The onset of Hopf bifurcations in rotatory thermal hydrodynamics is studied and the Hopf bifurcation number (threshold that the Taylor number crosses at the onset of Hopf bifurcations) is obtained.

Appendix

1.1 Proof of Property 1

Let $Re \lambda _i<0,\,\forall i$. In view of

$$\begin{aligned} \left\{ \begin{array}{l} \lambda _i<0\,\Leftrightarrow \, \lambda -\lambda _i=\lambda +|\lambda _i|,\\ \\ \{\lambda _i<0, \lambda _j<0\}\Rightarrow (\lambda -\lambda _i)(\lambda -\lambda _j)=\lambda ^2+|\lambda _i+\lambda _j|\lambda +|\lambda _i\lambda _j|,\\ \\ \{\lambda _r=-\alpha \pm i\beta , \alpha >0\}\Rightarrow (\lambda -\lambda _r)(\lambda -{{\bar{\lambda }}}_r)=(\lambda +\alpha )^2+\beta ^2,\\ \end{array} \right. \end{aligned}$$

and the Berout’s factorization

$$\begin{aligned} P(\lambda )=\displaystyle \prod _{i=1}^n (\lambda -\lambda _i), \end{aligned}$$

(2.8) immediately follows. One easily realizes that (2.9) are only necessary. In fact $\lambda ^3+27=0$ admits the roots $\lambda _{1,2}=\displaystyle \frac{3}{2}(1\pm i\sqrt{3})$ with $Re \lambda _{1,2}=\displaystyle \frac{3}{2}$.

1.2 On Routh–Hurwitz criterion

In the case $n=1$, the spectrum equation is $\lambda +a_1=0$, i.e. $\lambda <0\Leftrightarrow a_1>0$ which is the H-stability condition. In the case $n=2$, the roots of the spectrum equation $\lambda ^2+a_1\lambda +a_2=0$, in view of $\{a_1=-(\lambda _1+\lambda _2), a_2=\lambda _1\lambda _2\}$, have negative real part iff $a_1>0,\,a_2>0$. Being $\varDelta _1=a_1,\varDelta _2=a_1a_2$, $\{a_1>0,a_2>0\}\Leftrightarrow \{\varDelta _1=a_1>0, \varDelta _2=a_1a_2>0\}$. By induction one easily shows that the Hurwitz criterion holds for $n=3$. Let $\lambda _1$ be the real root in the case $n=3$. Then the spectrum equation can be written

$$\begin{aligned} P(\lambda , n=3)= & {} (\lambda -\lambda _1) P(\lambda , n=2)=(\lambda -\lambda _1)(\lambda ^2+a_1\lambda +a_2)=\\= & {} \lambda ^3+(a_1-\lambda _1)\lambda ^2+(a_2-a_1\lambda )\lambda -\lambda _1a_2=0 \end{aligned}$$

and the H-matrix and the H-conditions are

$$\begin{aligned} \left\| \begin{array}{c@{\quad }c@{\quad }c} a_1-\lambda &{}-\lambda _1a_2&{}0\\ 1&{}a_2-a_1\lambda _1&{}0\\ 0&{}a_1-\lambda _1&{}-\lambda _1 a_2 \end{array} \right\| ,\quad \left\{ \begin{array}{l} a_1-\lambda _1>0,\,\,-\lambda _1a_2>0,\\ a_1a_2-\lambda _1a^2_1+\lambda ^2_1a_1>0 \end{array}\right. \end{aligned}$$

(11.1)

Therefore, assuming that $P(\lambda ,n=2)$ verifies the H-conditions $\{a_1>0,a_1a_2>0\}$, guaranteeing $Re\lambda _{2,3}<0$, it follows that, $\lambda _1<0$ if and only if (11.1) occurs. An analogous procedure can be applied to any ADS constituted by odd number of equations

$$\begin{aligned} P(\lambda , n=1+2q)=(\lambda -\lambda _1)P(\lambda , n=2q)=0 \end{aligned}$$

with $\lambda _1$ real and $q\in {\mathbb {N}}$ and assuming that the criterion holds in the case $n=2q$. We refer to [2,3,4] for further details on the RH criterion and its elaborate proof $\forall n$.

1.3 Real eigenvalues of symmetric matrices

Let

$$\begin{aligned} \displaystyle \frac{d{\mathbf {u}}}{dt}=L{\mathbf {u}},\quad L=\left\| a_{ij}\right\| \end{aligned}$$

(11.2)

and let, by contradiction, $\lambda =\alpha +i\beta$ with $\beta \ne 0$ be a complex eigenvalue. In view of

$$\begin{aligned} \left\{ \begin{array}{l} L\cdot {\mathbf {k}}=\lambda I\cdot {\mathbf {k}},\\ \bar{{\mathbf {k}}}\cdot L\cdot {\mathbf {k}}=\lambda \bar{{\mathbf {k}}}\cdot I\cdot {\mathbf {k}}, \end{array}\right. \end{aligned}$$

(11.3)

with ${\mathbf {k}}=(k_1,k_2, \cdots , k_n)$ and $\bar{{\mathbf {k}}}=({\bar{k}}_1,{\bar{k}}_2, \cdots ,{\bar{k}}_n)$ complex conjugate eigenvectors

$$\begin{aligned} k_r=a_r+i b_r,\quad {\bar{k}}_r=a_r-i b_r, \end{aligned}$$

(11.4)

one has

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \sum _{r,s}^{1-n} a_{rs}k_s{\bar{k}}_r=\lambda \displaystyle \sum _{r,s}^{1-n}\delta _{rs} k_s{\bar{k}}_r,\\ \displaystyle \sum _{r,s}^{1-n} a_{sr}k_r{\bar{k}}_s=\lambda \displaystyle \sum _{r,s}^{1-n}\delta _{sr} k_r{\bar{k}}_s. \end{array}\right. \end{aligned}$$

(11.5)

In view of $a_{ij}=a_{ji}$ it follows that

$$\begin{aligned} \displaystyle \sum _{r,s}^{1-n} a_{rs}(k_s{\bar{k}}_r+k_r{\bar{k}}_s)=\lambda \displaystyle \sum _{r,s}^{1-n}\delta _{rs} (k_s{\bar{k}}_r+k_r{\bar{k}}_s). \end{aligned}$$

(11.6)

In view of

$$\begin{aligned} k_s{\bar{k}}_r+k_r{\bar{k}}_s=2(a_ra_s+b_rb_s), \end{aligned}$$

(11.7)

(11.6) implies

$$\begin{aligned} \lambda =\displaystyle \frac{\displaystyle \sum _{rs}^{1-n} a_{rs}(a_ra_s+b_rb_s)}{\displaystyle \sum _{r=1}^n(a^2_r+b^2_r)}= \text{ real } \text{ number } \Rightarrow \beta =0. \end{aligned}$$

(11.8)

1.4 Invariance principle

The spectrum of (11.2) is invariant with respect to the non-singular transformation

$$\begin{aligned} {\mathbf {u}}={\tilde{L}}\cdot {\mathbf {v}},\qquad (\det {\tilde{L}}\ne 0). \end{aligned}$$

(11.9)

In fact, (11.2) implies

$$\begin{aligned} {\tilde{L}}\displaystyle \frac{d{\mathbf {v}}}{dt}=(L{\tilde{L}}){\mathbf {v}} \end{aligned}$$

(11.10)

and hence

$$\begin{aligned} \displaystyle \frac{d{\mathbf {v}}}{dt}=(({\tilde{L}})^{-1}L{\tilde{L}}){\mathbf {v}}. \end{aligned}$$

(11.11)

The spectrum equation of (11.11) is

$$\begin{aligned} \det (({\tilde{L}})^{-1}L{\tilde{L}}-\lambda {\mathbf {I}})=0, \end{aligned}$$

(11.12)

with

$$\begin{aligned} I_{ij}=\delta _{ij}=\left\{ \begin{array}{c@{\quad }c} 1&{}i=j\\ 0&{}i\ne j. \end{array}\right. \end{aligned}$$

(11.13)

In view of $({\tilde{L}})^{-1}{\mathbf {I}}{\tilde{L}}={\mathbf {I}}$, one obtains

$$\begin{aligned} ({\tilde{L}})^{-1}L{\tilde{L}}-\lambda ({\tilde{L}})^{-1}{\mathbf {I}}{\tilde{L}}=({\tilde{L}})^{-1}(L-\lambda {\mathbf {I}}){\tilde{L}} \end{aligned}$$

(11.14)

and (11.12) is equivalent to

$$\begin{aligned} \det [({\tilde{L}})^{-1}(L-\lambda {\mathbf {I}}){\tilde{L}}]=\det ({\tilde{L}})^{-1}\det (L-\lambda {\mathbf {I}})\det {\tilde{L}}=0. \end{aligned}$$

(11.15)

Then $\det {\tilde{L}}\ne 0\Rightarrow \det ({\tilde{L}})^{-1}\ne 0$ and (11.15) reduces to $\det (L-\lambda {\mathbf {I}})=0$, the spectrum of (11.2).